Superman goes around the world in $2.5$ hrs, and Flash in $1.5$ hrs. How many times will they pass each other going in opposite directions in 24 hrs?

Since they run in opposite directions, from Superman's point of view, he's not moving and the Flash is running around the Earth at the speed $(1/2.5 + 1/1.5)$ laps /hr $= 16/15$ laps /hr. Multiplying this by the number of hours means Flash passes him $25.6$ times.

They meet when their lap length is $1$. Since they start at the same time, let $t$ be the time passed till they meet. So,

$$\frac{\text{Earth}}{2.5 h}t + \frac{\text{Earth}}{1.5 h}t = \text{Earth}$$ Solving this you get

$$\frac {4 \times \text{Earth}}{3.75 h}t = \text{Earth}$$ Solving, you get $t = 0.9375$ and $0.9375$ = $\frac{15}{16}$. Now, just divide $24$ by $\frac{15}{16}$ and you get $\frac{24 \times 16}{15}$ which is roughly $25.6$ and $25.6$ ~ $25$. So they meet $25$ times in an hour

Here is one way to look at it -

For them to complete one lap together, the time taken is

$t = \displaystyle \frac{2.5 \times 1.5}{4} = 0.9375 \ hr \tag1\ $

To derive $(1)$,

Speed of Flash $s_a = \frac{1}{1.5}$ lap / hr.
Speed of Superman $s_b = \frac{1}{2.5}$ lap / hr.

It is easy to see equation $(1)$ using relative speed. As they are traveling in opposite direction, to complete $1$ lap together, it will take them $\displaystyle \frac{1}{s_a + s_b}$ hr.

You can also derive equation $(1)$ as below -

When they meet first time, if Flash has traveled $x$, Superman has traveled $(1-x)$ in the same amount of time $t$.

$\displaystyle t = \frac{x}{s_a} = \frac{1-x}{s_b}$ and you can solve to get equation $(1)$.

Now for them to meet next, they again need to complete one lap together. So number of times they meet in $24$ hour is given by,

$ \displaystyle n = \lfloor \frac{24}{0.9375} \rfloor = 25$